Derivative of \( \displaystyle \sin^{4 x - 3}{\left(4 x - 3 \right)} \)
Problem 2.834 · hard
Differentiate \( \displaystyle f(x) = \sin^{4 x - 3}{\left(4 x - 3 \right)} \).
- \[ \frac{d}{d x} \sin^{4 x - 3}{\left(4 x - 3 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]rewriteRewrite the power using the exponential identity.≈ Checked numerically
- \[ = e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \frac{d}{d x} \left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(\sin{\left(4 x - 3 \right)} \right)} + \ln{\left(\sin{\left(4 x - 3 \right)} \right)} \frac{d}{d x} \left(4 x - 3\right)\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(\sin{\left(4 x - 3 \right)} \right)} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\frac{\left(4 x - 3\right) \frac{d}{d x} \sin{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{4 \left(4 x - 3\right) \cos{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) e^{\left(4 x - 3\right) \ln{\left(\sin{\left(4 x - 3 \right)} \right)}} \]derivativeDifferentiate the sine function.✓ Proved
- \[ = \left(\frac{\left(16 x - 12\right) \cos{\left(4 x - 3 \right)}}{\sin{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]algebraSubstitute the original function back for the exponential term.≈ Checked numerically
- \[ = \left(\left(16 x - 12\right) \cot{\left(4 x - 3 \right)} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]simplifySimplify the expression using the cotangent identity.≈ Checked numerically
- \[ = 4 \left(\left(4 x - 3\right) \cot{\left(4 x - 3 \right)} + \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \]simplifyFactor out the common constant 4.✓ Proved
Answer \( \left(\frac{16 x - 12}{\tan{\left(4 x - 3 \right)}} + 4 \ln{\left(\sin{\left(4 x - 3 \right)} \right)}\right) \sin^{4 x - 3}{\left(4 x - 3 \right)} \)
✓ Nihil obstat Lines: 8 proved, 3 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cos(4*x - 3) - log(sin(4*x - 3))*sin(4*x - 3))*exp((4*x - 3)*log(sin(4*x - 3))) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cos(4*x - 3) - log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*exp((4*x - 3)*log(sin(4*x - 3))))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*(((3 - 4*x)*cot(4*x - 3) - log(sin(4*x - 3)))*sin(4*x - 3)**(4*x - 2) + ((4*x - 3)*cos(4*x - 3) + log(sin(4*x - 3))*sin(4*x - 3))*sin(4*x - 3)**(4*x - 3))/sin(4*x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(4*x - 3) = 0 cot has poles at multiples of pi |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments cot has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(4*x - 3) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies logarithmic differentiation via rewriting as an exponential. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies logarithmic differentiation via rewriting as an exponential. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies logarithmic differentiation via exponential rewriting. Each step isolates a single rule application (rewrite, chain, product, derivative, simplify) and uses labels from the allowed vocabulary. The final answer matches the stated answer.gpt-oss:20b: pass 2026-09-26
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.